AcademyVectors In Rn
Academy
Three By Three Systems
Level 1 - Linear Algebra topic page in Vectors In Rn.
Principle
A \(3\) by \(3\) linear system is a system of three linear equations in three unknowns. Its rows describe planes in \(\mathbb R^3\), and its coefficient columns can be treated as vectors. Vector methods help identify whether the system has a unique solution.
Notation
\(x,y,z\)
unknown real numbers
\(A\)
the coefficient matrix of the system
\(\mathbf x\)
the vector of unknowns
\(\mathbf b\)
the right-hand side vector
\(\det A\)
the determinant of the coefficient matrix
\(\mathbf a_1,\mathbf a_2,\mathbf a_3\)
the column vectors of A
The Core Test
Write the system as a matrix equation:
Matrix equation
\[A\mathbf x=\mathbf b\]
For a \(3\) by \(3\) system, the coefficient matrix has three column vectors:
Column form
\[x\mathbf a_1+y\mathbf a_2+z\mathbf a_3=\mathbf b\]
The determinant gives the unique-solution test:
Unique solution test
\[\det A\ne 0\Longleftrightarrow \text{the system has a unique solution for every }\mathbf b\]
For columns \(\mathbf a_1,\mathbf a_2,\mathbf a_3\), the determinant is the scalar triple product:
Columns and volume
\[\det A=\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3)\]
If this volume is zero, the columns are coplanar and cannot span all of \(\mathbb R^3\).
Worked Cases
Question
Write the system \(x+2y-z=4\), \(3x-y+z=0\), \(2x+z=5\) in the form \(A\mathbf x=\mathbf b\).
Answer
Collect the coefficients of \(x\), \(y\), and \(z\) in each row. The coefficient matrix is \(A=\begin{pmatrix}1&2&-1\\3&-1&1\\2&0&1\end{pmatrix}\). The unknown vector is \(\mathbf x=\begin{pmatrix}x\\y\\z\end{pmatrix}\), and the right-hand side is \(\mathbf b=\begin{pmatrix}4\\0\\5\end{pmatrix}\). Therefore \(A\mathbf x=\mathbf b\).
Examples
Question
What does \(x\mathbf a_1+y\mathbf a_2+z\mathbf a_3=\mathbf b\) mean geometrically?
Answer
It means \(\mathbf b\) is being built as a linear combination of the three coefficient columns. The unknowns \(x\), \(y\), and \(z\) are the weights on those columns. Solving the system means finding the weights that produce \(\mathbf b\).